Multiplication table

9	8	7	6	5	4	3	2	�
18	16	14	12	10	8	6	4	2
27	24	21	18	15	12	9	6	3
36	32	28	24	20	16	12	8	4
45	40	35	30	25	20	15	10	5
54	48	42	36	30	24	18	12	6
63	56	49	42	35	28	21	14	7
72	64	56	48	40	32	24	16	8
81	72	63	54	45	36	27	18	9

A multiplication table is used to define a 'multiplication' operation for an algebraic system. Multiplication tables as they are used to teach schoolchildren multiplication are a grid where rows and columns are headed by the numbers to multiply, and the entry in each cell is the product of the column and row headings.

This table does not give the ones and zeros. That is because:

Anything times zero is zero.
Anything times one is itself. For example, 5�1=5.

Multiplication tables can define 'multiplication' operations for groups, fields, rings, and other algebraic systems.

The following table is an example of a multiplication table for the unit octonions (see octonion, from which this example is taken).

· 1 e₁ e₂ e₃ e₄ e₅ e₆ e₇
1 1 e₁ e₂ e₃ e₄ e₅ e₆ e₇
e₁ e₁ -1 e₄ e₇ -e₂ e₆ -e₅ -e₃
e₂ e₂ -e₄ -1 e₅ e₁ -e₃ e₇ -e₆
e₃ e₃ -e₇ -e₅ -1 e₆ e₂ -e₄ e₁
e₄ e₄ e₂ -e₁ -e₆ -1 e₇ e₃ -e₅
e₅ e₅ -e₆ e₃ -e₂ -e₇ -1 e₁ e₄
e₆ e₆ e₅ -e₇ e₄ -e₃ -e₁ -1 e₂
e₇ e₇ e₃ e₆ -e₁ e₅ -e₄ -e₂ -1

9	8	7	6	5	4	3	2	�
18	16	14	12	10	8	6	4	2
27	24	21	18	15	12	9	6	3
36	32	28	24	20	16	12	8	4
45	40	35	30	25	20	15	10	5
54	48	42	36	30	24	18	12	6
63	56	49	42	35	28	21	14	7
72	64	56	48	40	32	24	16	8
81	72	63	54	45	36	27	18	9

·	1	e₁	e₂	e₃	e₄	e₅	e₆	e₇
1	1	e₁	e₂	e₃	e₄	e₅	e₆	e₇
e₁	e₁	-1	e₄	e₇	-e₂	e₆	-e₅	-e₃
e₂	e₂	-e₄	-1	e₅	e₁	-e₃	e₇	-e₆
e₃	e₃	-e₇	-e₅	-1	e₆	e₂	-e₄	e₁
e₄	e₄	e₂	-e₁	-e₆	-1	e₇	e₃	-e₅
e₅	e₅	-e₆	e₃	-e₂	-e₇	-1	e₁	e₄
e₆	e₆	e₅	-e₇	e₄	-e₃	-e₁	-1	e₂
e₇	e₇	e₃	e₆	-e₁	e₅	-e₄	-e₂	-1

9	8	7	6	5	4	3	2	�
18	16	14	12	10	8	6	4	2
27	24	21	18	15	12	9	6	3
36	32	28	24	20	16	12	8	4
45	40	35	30	25	20	15	10	5
54	48	42	36	30	24	18	12	6
63	56	49	42	35	28	21	14	7
72	64	56	48	40	32	24	16	8
81	72	63	54	45	36	27	18	9

9	8	7	6	5	4	3	2	�
18	16	14	12	10	8	6	4	2
27	24	21	18	15	12	9	6	3
36	32	28	24	20	16	12	8	4
45	40	35	30	25	20	15	10	5
54	48	42	36	30	24	18	12	6
63	56	49	42	35	28	21	14	7
72	64	56	48	40	32	24	16	8
81	72	63	54	45	36	27	18	9